UML CS
Algorithms 91.503 (section 201)
Fall, 2006
Homework #8
Assigned: Wednesday, 11/15
Due: Wednesday, 11/29 at start of lecture
This assignment covers material on Chapter 29, Linear Programming.
1. (25 points) Given the following problem: A cargo plane has 3 compartments for
storing cargo: front, center and rear. These compartments have the following limits on
both weight and space:
Compartment
Front
Center
Rear
Weight Capacity (tons)
10
16
8
Space Capacity (cubic meters)
6800
8700
5300
The weight of the cargo in the respective compartments must be the same proportion
of that compartment’s weight capacity in order to maintain the balance of the plane.
The following 4 cargos are available for shipment on the next flight:
Cargo
C1
C2
C3
C4
Weight (tons)
18
15
23
12
Volume (cubic meters/ton) Profit (dollars/ton)
480
310
650
380
580
350
390
285
Fractional amounts of cargos can be accepted. Determine how much of each cargo
should be accepted and how to distribute them among the compartments so that the
total profit of the flight is maximized.
Formulate (but do not solve) a linear program for this problem.
Provide:
a)
b)
c)
d)
Objective function
Objective function sense:
maximize
or
minimize (circle one)
Constraints
Variables: for each variable, list its meaning and provide bounds
2. (25 points) An Integer Linear Program is a linear program in which some or all of the
variables are restricted to only have integer values.
Formulate (but do not solve) an Integer Linear Program for the 0-1 knapsack
problem, which is stated as follows (as in Cormen et al.):
A thief robbing a store finds n items: the i th item is
worth vi dollars and weighs wi pounds, where vi and wi
are integers. He wants to take as valuable a load as
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UML CS
Algorithms 91.503 (section 201)
Fall, 2006
possible, but he can carry at most W pounds in his
knapsack for some integer W. Which items should he
take?
Provide:
a)
b)
c)
d)
Objective function
Objective function sense:
maximize
or
minimize (circle one)
Constraints
Variables: for each variable, list its meaning and provide bounds
3. (50 points) Consider the following linear programming model:
Maximize
Subject to
y
y  x3
y   x  15
y   x / 5  3
a) Put the formulation into standard form.
b) Convert your answer to (a) into slack form.
c) Form the dual of your answer to (b).
d) Circle
i.
ii.
iii.
iv.
TRUE or FALSE for each statement below & briefly justify your choice.
The feasible region is bounded.
TRUE
FALSE
The feasible region is empty.
TRUE
FALSE
The feasible region is convex.
TRUE
FALSE
The objective function’s optimal value is bounded. TRUE
FALSE
e) Solve your linear program graphically (see attached grid). What is the optimal
value of your objective function? What are the values of your variables for this
optimal value?
f) Solve your linear program using SIMPLEX. What is the optimal value of your
objective function? What are the values of your variables for this optimal value?
Show the linear program resulting from each iteration of SIMPLEX.
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UML CS
Algorithms 91.503 (section 201)
3 of 3
Fall, 2006